Find the area of the triangle formed by the lines joining the vertex of the parabola `x^(2) = 8y` to the ends of its latus rectum.

Options

A. (a) 8 sq. units
B. (b) 16 sq. units
C. (c) 4 sq. units
D. (d) 32 sq. units

Answer

To find the area of the triangle formed by the lines joining the vertex of the parabola \( x^2 = 8y \) to the ends of its latus rectum, we will follow these steps:

### Step 1: Identify the vertex and focus of the parabola
The given parabola is \( x^2 = 8y \). This can be compared with the standard form of a parabola \( x^2 = 4ay \).

From the equation \( 4a = 8 \), we can find \( a \):
\[ a = \frac{8}{4} = 2 \]
Thus, the vertex \( O \) of the parabola is at the origin \( (0, 0) \) and the focus \( S \) is at \( (0, a) = (0, 2) \).

**Hint:** Remember that the vertex of the parabola is the point where it opens from, and the focus is a point inside the parabola that helps define its shape.

### Step 2: Determine the ends of the latus rectum
The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry that passes through the focus. The endpoints of the latus rectum can be found using the coordinates:
\[ A = (-2a, a) \quad \text{and} \quad B = (2a, a) \]
Substituting \( a = 2 \):
\[ A = (-2 \cdot 2, 2) = (-4, 2) \quad \text{and} \quad B = (2 \cdot 2, 2) = (4, 2) \]

**Hint:** The ends of the latus rectum are symmetric about the y-axis and have the same y-coordinate as the focus.

### Step 3: Set up the coordinates of the triangle
The triangle is formed by the points \( O(0, 0) \), \( A(-4, 2) \), and \( B(4, 2) \).

**Hint:** Identify the vertices of the triangle clearly; they are crucial for calculating the area.

### Step 4: Use the formula for the area of a triangle
The area \( A \) of a triangle formed by points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
- \( O(0, 0) \) gives \( (x_1, y_1) = (0, 0) \)
- \( A(-4, 2) \) gives \( (x_2, y_2) = (-4, 2) \)
- \( B(4, 2) \) gives \( (x_3, y_3) = (4, 2) \)

Substituting these values into the area formula:
\[ A = \frac{1}{2} \left| 0(2 - 2) + (-4)(2 - 0) + 4(0 - 2) \right| \]
This simplifies to:
\[ A = \frac{1}{2} \left| 0 - 8 - 8 \right| = \frac{1}{2} \left| -16 \right| = \frac{16}{2} = 8 \]

### Conclusion
The area of the triangle formed by the vertex of the parabola and the ends of its latus rectum is \( 8 \) square units.

**Final Answer:** The area of the triangle is \( 8 \) square units.

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